Pioneering theoretical developments in the field of plasticity in the middle of the last century led to the development of simple yet practical and powerful analytical tools that were used to rapidly estimate the limit loads of simple bodies and structures. However, engineers increasingly demand more generally applicable methods, which normally require the use of numerical methods rather than analytical methods. Unfortunately, and despite their potential usefulness in practice, relatively few generally applicable numerical methods have to date been explored for this application. Of the methods that have been explored, the finite element method has proved the most popular with researchers over the past few decades as can be appreciated from, for example, Hodge, P. & Belytschko, T. 1968 Numerical methods for the limit analysis of plates. J App. Mech. 35 (4), 796-802, Lysmer, J. 1970 Limit analysis of plane problems in soil mechanics. Journal of the Soil Mechanics and Foundations Division ASCE 96 (4), 1311-1334; Sloan, S. 1988 Lower bound limit analysis using finite elements and linear programming. Int. J. Num. Anal. Meth. in Geomech. 12 (4), 61-77; and Makrodimopoulos, A. & Martin, C. 2006 Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Num. Meth. in Eng. 6 (4), 604-634, which are all incorporated herein in their entirety by reference for all purposes. However, at the time of writing finite element limit analysis has still to find widespread usage in engineering practice, with engineers instead being forced to rely either on analytical methods of limited applicability or on potentially cumbersome iterative elastic-plastic analysis methods.
Modern finite element limit analysis formulations, typically, involve discretisation of a body using both solid and interface elements, the latter being placed between solid elements to permit jumps in the stress or strain rate fields. Such formulations may, therefore, be considered as hybrid continuous-discontinuous analysis methods. Suitable finite element shape functions are used to ensure that the internal stresses in the solid elements satisfy specified yield criteria (equilibrium formulation), or that the flow rule is satisfied throughout each solid element (kinematic formulation). The failure surfaces used in limit analysis are generally non-linear, although they may be linearised to permit the problem to be solved using linear programming (LP).
Alternatively, problems involving certain non-linear yield surfaces can be treated using efficient convex programming techniques. Efficient convex programming techniques include, for example, Second Order Cone Programming used by Makrodimopoulos & Martin mentioned above. Unfortunately, the solutions obtained using finite element limit analysis are often highly sensitive to the geometry of the original finite element mesh, particularly in the region of stress or velocity singularities. Although meshes may be tailored to suit the problem in hand this is clearly unsatisfactory since advance knowledge of the mode of response is required. Adaptive mesh refinement schemes can potentially overcome this problem as can be appreciated from, for example, Lyamin, A., Sloan, S., Krabbenhoft, K. & Hjiaj, M. 2005 Lower bound limit analysis with adaptive remeshing. Int. J. Numer. Meth. Engng. 63, 1961-1974, which is incorporated herein in its entirety for all purposes. However, the resulting analysis procedure is complex considering the simple rigid-plastic material idealisation involved.
Finite element analysis is traditionally concerned with formulation and solution of a continuum mechanics problem. Alternatively, it is possible to consider a potentially simpler discontinuous problem. This involves directly identifying the discontinuities that form at failure such as, for example, slip-lines that transform a planar continuum into discontinua. However, previous attempts to develop discontinuous limit analysis formulations have been largely unsuccessful, principally because previous formulations have permitted only a severely limited range of possible failure mechanisms to be considered. For example, Munro, J. & Da Fonseca, A. 1978 Yield line method by finite elements and linear programming, The Structural Engineer 56B (2), 37-44, which is incorporated herein in its entirety for all purposes, prescribed that discontinuities could only coincide with the boundaries of rigid elements, meaning, for example, that fan zones in failure mechanisms could not easily be identified (unless the mesh was pre-defined to take this form). Modern hybrid continuous-discontinuous finite element limit analysis formulations also prescribe that discontinuities can only coincide with element boundaries, although such formulations partly compensate for this by allowing displacements in the elements themselves.
It is an object of embodiments of the present invention to at least mitigate one or more of the problems of the prior art.